Knot theory and low-dimensional topology

打结理论是数学的一个分支，它研究的是两端融合在一起的打结弦的理想形式。核心问题是区分两个结什么时候相同，什么时候不同。这门学科的早期先驱们发展了启发式的方法，他们用这种方法把质数结编成10个交叉点的表格。只有引入了严格的数学技术，这门学科才有了坚实的数学基础。有了更复杂的技术和工具，数学家们现在已经把结列成表格，多达16个交叉点。低维拓扑包括对2、3和4维流形的研究。中心问题是对三维和四维流形进行分类。流形是一种几何形状，在任何一点附近看起来都像欧几里得空间一样平坦，但其整体结构可能是扭曲和弯曲的。球的表面和甜甜圈的表面提供了具体的例子；想象一个沙滩球或内胎。两者都不是平的，然而，球或内管的任何破裂都可以用一个小矩形补丁和一点胶水来修复。由于流形是局部不可区分的，数学家寻找反映流形的全局弯曲和扭曲的不变量。例如，想象一只近视眼的昆虫生活在一个球的表面或一张平的纸上。它怎么能把两者区分开来呢？一种方法是计算欧拉特征V - E + F，即顶点数减去边数加上三角剖分中的面数，这与三角剖分无关，是拓扑不变量的一个例子。这两个数学分支密切相关。其中一个原因是三维和四维流形可以沿着结点或链接构造成Dehn手术。结理论和低维拓扑在物理、化学和生物学中有着广泛的应用。申请人建议引入新的三维流形不变量及其内部结点。他将开发计算这些不变量的新方法。不变量将定义和研究使用代数和几何方法，包括规范理论和量子拓扑。有许多有趣的学生研究项目与这些问题相关，申请人将促进来自不同群体的学生和博士后的参与。他将营造一个包容的研究环境，鼓励女性和少数族裔的学习和进步。这项研究计划的长期效益是双重的：所获得的知识将有助于确定几何方法在多大程度上可以提供新的结点和3流形不变量，培训计划将培养各级高素质人才，具有必要的分析和计算技能，在信息化经济中担任领导角色。

Knot theory is a branch of mathematics that studies an idealized version of a knotted string where the two ends are fused together. The central problem is to distinguish when two knots are the same and when they are different. The early pioneers of the subject developed heuristic methods which they used to tabulate prime knots to 10 crossings. It was only with the introduction of rigorous mathematical techniques that the subject was placed on a solid mathematical footing. With many more sophisticated techniques and tools, mathematicians have now tabulated knots up to 16 crossings.   Low-dimensional topology involves the study of manifolds in dimensions 2,3 and 4. The central problem is to classify 3 and 4-dimensional manifolds. Manifolds are geometric shapes that, near any point, look flat like Euclidean space but whose global structure may be twisted and curved. The surface of a ball and the surface of a doughnut provide concrete examples; imagine a beach ball or an inner tube. Neither is flat, nevertheless, any rupture to the ball or inner tube can be repaired with a small rectangular patch and a bit of glue. Since manifolds are locally indistinguishable, mathematicians search for invariants that reflect the manifold's global curving and twisting. For example, imagine a near-sighted insect living on either the surface of a ball or a flat sheet of paper. How could it tell the two apart? One method would be to compute the Euler characteristic V - E + F, the number of vertices minus the number of edges plus the number of faces in a triangulation, which is independent of the triangulation and is an example of a topological invariant. These two branches of mathematics are intimately related. One reason is that 3 and 4-dimensional manifolds can be constructed as Dehn surgeries along knots or links. Knot theory and low-dimensional topology have numerous applications to physics, chemistry and biology. The applicant proposes to introduce new invariants of 3-dimensional manifolds and knots inside them. He will develop new approaches for computing these invariants. The invariants will be defined and studied using algebraic and geometric methods, including gauge theory and quantum topology. There are many interesting student research projects related to these questions, and the applicant will promote participation from a diverse group of students and postdoctoral fellows. He will foster an inclusive research environment that encourages learning and advancement of women and underrepresented minorities. The long-term benefits of this research program are two-fold: the knowledge gained will help determine to what extent geometric methods can deliver new invariants of knots and 3-manifolds, and the training program will produce highly qualified personnel at all levels, with the necessary analytical and computational skills to take leadership roles in the information-based economy.